Questions tagged [fundamental-solution]

Questions on fundamental solutions of an ordinary differential equation.

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37 views

Separate variables and use partial fractions with IVP

Solve $\frac{dx}{dt} = 3x(x-5)$ This is all done in terms of $x(t)$: $x(0)=8$ $\frac{dx}{dt} = 3x(x-5)$ $\int \frac{dx}{3x(x-5)} = \int dt$ LHS: $\frac{A}{3x} + \frac{B}{x-5}$ $1 = Ax - 5A + ...
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2answers
27 views

Solution verification on homework problem. Separable first order ODE IVP.

The answer is supposedly $y^2 = 1 + \sqrt{x^2 - 16}$ I don't know where I went wrong cause I know for a fact that my substitution of $x = 4 \sec(\theta)$ is correct. I know for a fact that after ...
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0answers
44 views

fundamental solution of an ODE of second order

Let us consider the following homogenous ODE of second order: $$x''(t)+a_1(t)x'(t)+a_2(t)x(t)=0$$ where $a_1$ and $a_2$ are continuous functions. Are there conditions on these functions such that one ...
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65 views

Fourier transform of the Hankel function of first kind and zero order

Given the Hankel function of first kind and zero order $$ H_0^{(1)}(\omega|x-y|), $$ with $|x-y| \geq C >0$, I would like to calculate its Fourier transform, that is $$ \int_{-\infty}^{\infty} ...
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39 views

I have a question about Elzaki variational iteration method.

How to get from the first line to the second line? enter image description here $$u_{n+1}(x,t) = u_n (x,t) - \int_0^t \left\{ (u_n)_s(x,s) - \frac{\partial}{\partial s} G(x,s) + \frac{\partial}{\...
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1answer
56 views

Explicit formula of a fundamental matrix

I'm trying to solve the following differential system explicitly $$\left( \begin{array}{c} y \\ z% \end{array}% \right) ^{\prime }=\left( \begin{array}{cc} 0 & b(t) \\ \delta b(t) & a(t)% \...
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1answer
39 views

Why $ \int_{0}^{t }\left(\frac{\partial}{\partial s} f(x,s)\right)ds =f(x,t)$ [closed]

Why is the following true? $$\int_{0}^{t }\left(\frac{\partial}{\partial s} f(x,s)\right)ds =f(x,t)$$ Can you give any hints?
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1answer
67 views

Solution of PDE (heat equation) unique? [closed]

I am new to PDEs and need some help. In fact, I am not studying PDEs but I need the following for another result not related to this topic. Consider the initial value problem: $\frac{d}{dt}w(t,x) - \...
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0answers
32 views

Integrating log |z-w| over a disk in the complex plane.

Let $ E=\frac{1}{\pi} \log |z|^{2} $ and $ \chi_{r}=\frac{1}{\pi r^{2}} \chi_{\Delta(0, r)} $ where $\chi_{\Delta(0, r)}$ is the characteristic function of the disk with radius $r$ centered at $0$, ...
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4answers
75 views

Differential equation $y'' - y + 2\sin(x)=0$

I need help and explanation with this differential equation. Actually I really don't know how to solve just this type of equations. So the problem: $$y'' - y + 2\sin(x)=0$$ In my opinion first of all ...
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32 views

Transition densities for (elliptic) SDEs [or: Fundamental solutions for (elliptic) PDEs]

General question: When are the transition kernels corresponding to the SDE $$ dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \quad t\in[0,1], \tag{$\heartsuit$}$$ absolutely continuous w.r.t. Lebesgue's ...
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1answer
33 views

Fundamental Theorem of Calculus Piecewise Function help?

Let: $$f(x)= \begin{cases} 0, & x < -4 \\ 5, \qquad\quad& \llap{-4 \le{}} x < -1 \\ -2, & \llap{-1 \le{}} x < 3 \\ 0,& x \ge 3\end{cases}$$ $$g(x) = \int_{-4}^x f(t)...
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2answers
68 views

On the Schrodinger fundamental solution

Let $e^{it\Delta}$ be the fundamental Schrodinger solution. If $u_0$ is the corresponding initial data to the problem associated to Schrodinger free equation $u_t = i\Delta u$ and $S(\mathbb{R}^N)$ ...
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0answers
64 views

The fundamental solution for Laplace's equation in cylindrical coordinates

I am working my way through the excellent textbook of Garabedian on partial differential equations and have two questions related to the topic of the fundamental solution in chapter 5: Garabedian ...
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1answer
44 views

Find differential equation general solution

The task is to find $u_{xy}+yu_x+xu_y+xyu=0$ this equation general solutions. First of all I wrote it as $\frac{d}{dx}(u_y+yu)+x(u_y+yu)=0$ Then marked $u_y+yu=t$ and got equation $t_x+xt=0$ And ...
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2answers
40 views

Fundamental matrix of a particular system

I'm interested of finding a closed formula for the fundamental matrix to the system $$\eqalign{ & y'(t) = a(t)z(t) \cr & z'(t) = \delta a(t)y(t) \cr} $$ $$(y(0),z(0)) = ({y_0},{z_0})$$ ...
2
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0answers
27 views

Analytical Solution to a linear PDE with delta function like source term

I have a time-independent Vlasov equation with source term in $R^2$: $$v\cdot\partial{f}/\partial{x}+\partial{f}/\partial{v}=s(x,v)$$ where $s(0,0)=500$, else $s(x,v)=0$. And I was wondering how to ...
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1answer
43 views

Obtaining the Fundamental Solution of $1-\partial_x^2$

How does one obtain the fundamental solution of the differential operator $(1-\partial_x^2)$? There does not seem to be any easily accesible literature specifically describing how this is done, except ...
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0answers
44 views

Modified Helmholtz for imaginary constant

The Green's function for the free space modified Helmholtz equation in two dimensions is \begin{equation} \alpha^{2}u - \Delta u = 0, \end{equation} with $\alpha^{2}$ real, is $K_{0}$. However, I am ...
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1answer
61 views

Wronski-Test for linear ODE

To test solutions of linear ODEs for lineary independence you can determine the wronski determinant. The theorem says if you have a solution to the linear ODE in the form: $$ \dot{\vec{x}}(t) = A \...
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1answer
52 views

Approach of Variation of parameters (Variation of Constants)

i have a question about the approach "Variation of parameters (also known as variation of constants). Imagine we have non-homogene ODE of the form: $$ y' = a(x) \cdot y + b(x)$$ The homogene solution ...
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45 views

how to find the fundamental solution of $-\Delta u + e^u - e^{-u} = \delta(\vec{x})$ in 2D?

$-\Delta u + e^u - e^{-u} = \delta(\vec{x})$, where $\Delta$ is the Laplace operator and $\delta(\vec{x})$ is the Dirac's delta function and satisfies: $\delta(\vec{x}) = \begin{cases} 0, & \vec{...
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0answers
87 views

A problem about fundamental solution of Laplace equation

I would like to derive a solution to the following equation in $R^{3}$: $$-\Delta u(x) + e^{u(x)} - e^{-u(x)} = \delta(x)$$where $\Delta$ is the Laplace Operator and $\delta(x)$ is the Dirac's delta ...
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0answers
43 views

Hipoelliptic altered wave operator?

We know that the wave operator in $\mathbb{R}^{2}$: $L=\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}$ Can I say that $L-\lambda$, $\lambda(x,y) \in C^\infty(\mathbb{R}^2)$ is not ...
2
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1answer
67 views

Floquet theorem in mathematics and physics

I am using the notation from Proofwiki. The Floquet theorem states that for a continuous matrix function $A(t)$ with period $T$ and a fundamental matrix $\Phi(t)$ of the system $x'(t)=A(t)x(t)$, it is ...
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0answers
76 views

Fundamental Matrix of OED 3x3 Constant matrix

Given the constant matrix A= \begin{bmatrix}4&-1&0\\3&1&-1\\1&0&1\end{bmatrix} find the fundamental matrix for the system $y'=Ay$. After attempting to solve the system, I ...
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2answers
65 views

General solution of ODE of a constant $3\times 3$ matrix

Determine the general solution of the system $y'=Ay$ , where $A$ is a constant matrix, defined by $$A = \begin{pmatrix}-5&-8&4\\2&3&-2\\6&14&-5\end{pmatrix}$$ After ...
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0answers
66 views

Finding Eigenvector from a 3x3 matrix

Given the constant matrix A = \begin{bmatrix}4&-1&0\\3&1&-1\\1&0&1\end{bmatrix} Find the fundamental matrix. After finding the eigenvalue $\lambda$=2 with multiplicity 3, ...
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1answer
39 views

Finding Eigenvector from eigenvalue

Given the constant matrix \begin{bmatrix}-3&4&-2\\1&0&1\\6&-6&5\end{bmatrix} find the fundamental matrix. After, finding the corresponding eigenvalues which are $\lambda$ = 2,...
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66 views

Difference of fundamental solutions

I am reading a set of notes in which the author claims that the difference between any two fundamental solutions (to the Laplacian) is harmonic. That is, if $E_1$ and $E_2$ are fundamental solutions ...
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ODE, free-space Green's Function

In my PDE study, I learned about $\textit{free-space Green's Functions}$ and I thought if there was also for ODE ($\frac{d}{dx^2}$ or Sturm-Liouville). I looked for free-space green's functions for ...
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1answer
276 views

$N$-dimensional Heat equation + BC's

Problem I have to solve the nonhomogeneous classic problem $$\left(P_{1}\right)\;\,\left\{ \begin{aligned} u_{t}\;-\; \Delta u\; &= \;f& &\textrm{on}\;\;\; \Omega\times\left(0,\,\infty\...
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3answers
47 views

solve for second order linear differential equations [closed]

I got the sum of A is 0? There is no solution to this? Can someone please help. Thanks! $$y''-4y'+4y=-6e^{2t}$$
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2answers
44 views

Writing $1-e^{-xy}$ as a square.

Is it possible to write $1-e^{-xy} = r(x)r(y)$ for some function $r$ where $x,y$ are positive real numbers. I was just wondering to try to express that quantity like that. I tried solving the equation ...
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1answer
48 views

Different equation general solution

Hi I am looking for the general solution of the following problem: $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ Initial condition: $u(x,0) = -x $ and $u(0,t) = 0 $ and $u(1,...
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1answer
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Differential Equations: Which values of b = damping coefficient does the typical solution approach the equilibrium position most rapidly

Here's where I'm at with it: $\lambda ^2+b\lambda +3=0$ $\lambda _1=\frac{\left(-b+\sqrt{b^2-12}\right)}{2},\:\lambda \:_2=\frac{\left(-b-\sqrt{b^2-12}\right)}{2}$ I do not know how to complete this ...
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1answer
196 views

A question regarding the fundamental solution of a 1D Laplace equation

Disclaimer: My mathematical background is mathematical physics so this question most likely lacks rigor. I am trying to understand the nature of the singularity of the fundamental solution (aka Green ...
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0answers
76 views

calculating first and fundamental form coefficients with surface normal and Gaussian curvature

I have some points (point cloud), I need first and second fundamental form coefficients at each point. before I used some polynomial surface fitting algorithm then these coefficients were calculated....
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1answer
63 views

What is $\frac {\partial \Gamma} {\partial \nu}$ on $\partial B_{\rho} (y)$?

I am studying Laplace's equation from the book "Elliptic partial differential equations of second order" written by Gilbarg and Trudinger. Here I am struggling to grasp a concept regarding the ...
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1answer
281 views

How to use the Wronskian to find the in homogeneous solution of an ODE?

Suppose I have an ODE in the form: $ y'' + p(x)y' + q(x)y = f(x) $ Suppose now that I have obtained the solution to the equation $ y'' + p(x)y' + q(x)y = 0 $ which is in the form $ y(x) = c_1 ...
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2answers
143 views

Roots of linearly independent solutions of homogeneous ODE 2

If $y_1$ and $y_2$ are linearly independent solutions (set of fundamental solutions) of homogeneous ODE 2 $$ y'' + p(t) y' + q(t) y = 0 $$ prove that between 2 ...
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1answer
170 views

bessel function of second kind

I'm working through "Elementary Differential Equations", 10th Ed (Boyce/DiPrima) on my own -- 120 miles from the nearest easily accessible prof or grad student -- and I've a question on a technique in ...
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1answer
174 views

How can a point that makes the ODE undefined be included into a domain of any solution?

To illustrate the problem that I'm having, consider the following ODE: $$y' = \frac{y}{1+x}$$ To solve this ODE, my professor had done the followings: Assuming $y(x) \not = 0$, and $x \not = -1$ (...
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2answers
46 views

Real Fundamental- System/Matrix of a Differential equation

We consider: $$y''' - 2y'' + 2y' - y = 0$$ The real solution to this equation is: $$y(x) = c_3e^{x} + c_2e^{x/2}sin\left(\frac{\sqrt{3}x}{2}\right) + c_1e^{x/2}cos\left(\frac{\sqrt{3}x}{2}\right)$...
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0answers
142 views

Estimates for the derivatives of the solution of heat equation

Let $ u_0\in L^\infty (\mathbb{R^d})\cap C^\alpha(\mathbb{R}^d), \alpha \in (0,1), \Phi $ the fundamental solution of the heat equation and $u(t,\cdot ):= \Phi (t,\cdot )*u_0(\cdot)$ for $t>0$. Let ...
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1answer
41 views

Deriving an Explicit Formula using Fundamental Solutions

I am currently trying to find a Fundamental Solution to the Equation: $$\Delta\Phi+c\Phi=\delta_0$$ To find an explicit formula for $$\Delta u+cu=f$$ Now I know I am supposed to look for radially ...
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0answers
116 views

Finding the Fundamental Solution of the Operator $\frac{\partial}{\partial x}+c$

I am trying to find the Fundamental Solution for the Operator $\frac{\partial}{\partial x}+c$. I approach it by considering a radial function $v(r)$, and considering the ODE $v'(r)+c v=0$. This ODE ...
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1answer
1k views

2D Green's Function on a Disk with Radius $a$

I am currently trying to solve a problem, but am having a few difficulties. The question is to find the Green's Function of the Laplacian on the disk centered at the origin with radius $a$. Denote $\...
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1answer
20 views

CDA Fundamentals of Comp.S - Reduce an equation

how can I reduce: $co = ci'ab+cia'b+ciab'+ciab$ to: $co = ab + cia + cib$ I don't know how they did that on the book... This is for Adders, working with circuits. Thank you very much
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1answer
46 views

Trying out Green functions in simple context

I am a physics student, and have had to use Green's function methods prior (in electrodynamics for example), but things were always badly explained. Now I am trying to brush up on things a bit and ...